Gaussian Curvature, Normal Curvature, and the Shape Operator

1. The problem statement, all variables and given/known data
Let [tex]u_1, u_2[/tex] be orthonormal tangent vectors at a point p of M. What geometric information can be deduced from each of the following conditions on S at p?

a) [tex]S(u_1) \bullet u_2 = 0[/tex]

b) [tex]S(u_1) + S(u_2) = 0[/tex]

c) [tex]S(u_1) \times S(u_2) = 0[/tex]

d) [tex]S(u_1) \bullet S(u_2) = 0[/tex]

2. Relevant equations
If v and w are linearly independent tangent vectors at a point p of M, then [tex]S(v) \times S(w) = K(p)v \times w[/tex], where [tex]K= det S[/tex].

c) By the given formula, we know that [tex]K(p) = 0[/tex] since [tex]u_1 \times u_2 \neq 0[/tex]. But when [tex]K(p)=0[/tex], there are two cases, depending on the principal curvature, which I don't have any information about.